Scalar: single-valued quantities in measurement and mathematics
A scalar is a single numerical value that expresses magnitude only. Used in measurement, physics, mathematics and computing, it contrasts with vectors and matrices and appears in units, scalar fields and algebra.
A scalar is a quantity that can be completely described by a single number (often together with a unit). In ordinary language a scalar is what we mean by a plain number used to indicate size or amount. When we report a measurement we commonly give a scalar value and a unit: for example, the length of a rod might be "2 metres" or "30 centimetres", and that single pair of pieces of information suffices to convey the measurement.
Scalars in physical measurement
In physics and engineering a scalar is any measurable quantity that has magnitude but no direction. Typical examples are mass, luminous intensity, temperature, time and energy. Each of these is reported as a single value together with a unit: mass in kilograms, length in metres, luminous intensity in candelas, and so on. A scalar such as distance differs from a vector quantity such as displacement because the scalar does not record any direction, only magnitude.
Mathematical meaning
Mathematically a scalar is an element of the underlying field used in linear algebra and related areas. Commonly this field is the real numbers or the complex numbers; in that sense a scalar is simply a real or complex number that can serve as a coefficient. Scalars enter linear algebra through operations such as scalar multiplication, where a scalar multiplies every component of a vector or every entry of a matrix.
For example, multiplying a vector by a scalar changes its magnitude: a positive scalar scales length while a negative scalar also reverses direction. Multiplying a matrix by a scalar scales each entry and thus scales the linear transformation represented by the matrix. These operations make scalars central to the structure of vector spaces and matrix algebra.
Scalar fields and functions
A scalar field assigns a scalar value to every point of a region of space. Temperature distribution in a room, pressure in a fluid, and concentration of a chemical species in a solution are common examples of scalar fields. Unlike vector fields, which assign a vector to each point, scalar fields provide a single number per point and are widely used in physics, meteorology and numerical simulation because they are simpler to visualize and analyze in many contexts. Measurement overview and introductions in applied science describe how scalar fields are represented and sampled in practice.
Units, conventions and examples
Scalars almost always appear with units when used in measurement. Units provide the scale that interprets the numerical value: for instance, the same numeric scalar can represent very different magnitudes depending on whether the unit is metres, centimetres or kilometres. Common units associated with scalar quantities include metres for length, kilograms for mass, candelas for luminous intensity and seconds for time. In practical work it is important to report both the scalar and its unit to avoid ambiguity.
Distinctions and related notions
Scalars should be distinguished from vectors and matrices. A vector carries both magnitude and direction and is represented by an ordered list of scalars; a matrix arranges scalars in a rectangular array and can represent linear transformations. Many operations combine scalars with these objects: adding vectors requires vectors, but multiplying a vector or matrix by a scalar is a standard and well defined operation.
- Examples of scalar quantities: temperature, mass, time, energy, speed.
- Units: metres, kilograms, seconds, candelas; the unit must accompany the scalar in measurement contexts.
- Algebraic role: scalars form the coefficients in linear combinations and belong to the field underlying a vector space.
Applications and notes
Scalars appear across science, engineering and computing. In computer programming a scalar variable holds a single value (distinct from arrays or collections). In modelling and simulation scalar fields are discretized and used as inputs or outputs of numerical methods. When interpreting any reported value, check the context and the unit: the bare number without unit can be ambiguous. For further reading see elementary texts on measurement, introductory linear algebra, and applied guides to units and quantities. Related entries and resources include discussions of units and standards, comparisons of distance versus displacement, and definitions of photometric quantities. Additional overview material is available in resources on numbers, measurement, and instructional material for science and engineering education.
This article summarizes common uses and the mathematical role of scalars; for specialized contexts (relativistic physics, abstract algebra, computer data types) consult domain-specific sources where conventions and definitions may be refined.
Scalars in physics
In physics, scalars are used to describe physical quantities that are independent of direction. Examples of scalar physical quantities are the mass of a body, its temperature, its energy and also its distance from another body (as the magnitude of the difference of the position vectors). In other words, a scalar physical quantity does not change with changes in location or orientation. If, on the other hand, a direction is required for the complete description of the quantity, as in the case of the force or the velocity, a vector is used, and in the case of dependence on several directions, a tensor (more precisely: tensor of the 2nd level or even higher).
The velocity of a particle has the direction in which the particle is moving. Since the direction changes with rotations, the velocity is not a scalar, but a vector. But the magnitude of the velocity does not change with rotations and is a scalar.
Whether a quantity is a scalar depends on the transformation group under consideration. Thus energy is a scalar with respect to rotations, but in relativity it is a component of a four-vector.
A subgroup of the scalars are the pseudoscalars, which reverse the sign under a plane reflection.
Extensions and delimitation of similar terms
- Quadratic matrices, which (taken as a linear mapping of a vector space onto itself)
correspond to a multiplication of each vector by a fixed scalar λ to have the property scalar. They are diagonal matrices whose entries on the diagonal are all equal to λ .
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AlegsaOnline.com Scalar: single-valued quantities in measurement and mathematics Leandro Alegsa
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